The classification problem for coadjoint orbits for the action of symplectic (or area-preserving) diffeomorphisms in two dimensions was known to specialists in view of its application in fluid dynamics since the 1960s, and it was explicitly formulated in [1]} in 1998.
The same classification problem also arises in Poisson geometry since coadjoint orbits are symplectic leaves of the Lie-Poisson bracket, and also in representation theory in connection with the orbit method of A.Kirillov [2]}. In the recent work [3]} the orbit method was applied to the symplectomorphism group of the two-sphere. The classification of generic coadjoint orbits was obtained in [4]}, [5]} for the case of closed surfaces. In [4]} there is a list of open questions for the case of surfaces with boundary. We answer all those questions in Sections  and .
